data: simulated data for demonstrating the features of Blend

In Blend: Robust Bayesian Longitudinal Regularized Semiparametric Mixed Models

simulated data for demonstrating the features of Blend

Description

Simulated gene expression data for demonstrating the features of Blend.

Format

The data object consists of 8 components: y, x, t, J, kn and degree.

Details

The data and model setting

Consider a longitudinal study on n subjects with J_i repeated measurements for each subject. Let Y_{ij} be the measurement for the i-th subject at each time point t_{ij}, (1 \leq i \leq n, 1 \leq j \leq J_i). We use an m-dimensional vector X_{ij} to denote the genetic factors, where X_{ij} = (X_{ij1},...,X_{ijm})^\top. Z_{ij} is a 2 \times 1 covariate associated with random effects and \zeta_{i} is a 2 \times 1 vector of random effects corresponding to the random intercept and slope model. We have the following semi-parametric quantile mixed-effects model:

Y_{ij} = \alpha_0(t_{ij}) + \sum_{k=1}^{m} \beta_{k}(t_{ij}) X_{ijk} + Z_{ij}^\top \zeta_{i} + \epsilon_{ij}, \zeta_{i} \sim N(0, \Lambda)

where the fixed effects include: (a) the varying intercept \alpha_0(t_{ij}), and (b) the varying coefficients \beta(t_{ij}).

The varying intercept and the varying coefficients for the genetic factors can be further expressed as \alpha_0(t_{ij}) and \beta(t_{ij}) = (\beta_{1}(t_{ij}), ..., \beta_{m}(t_{ij}))^\top.

For the random intercept and slope model, Z_{ij}^\top = (1, j) and \zeta_{i} = (\zeta_{i1}, \zeta_{i2})^\top.

Furthermore, Z_{ij}^\top \zeta_{i} can be expressed as (b_i^\top \otimes Z^\top_{ij}) J_2 \delta, where \zeta_{i} = \Delta b_i, \Lambda = \Delta \Delta^\top, and

b_i^\top \otimes Z^\top_{ij} = (b_{i1} Z_{ij1}, b_{i1} Z_{ij2}, b_{i2}Z_{ij1}, b_{i2} Z_{ij2})^\top.

In the simulated data,

Y = \alpha_{0}(t)+\beta_{1}(t)X_{1} + \beta_{2}(t)X_{2} + \beta_{3}(t)X_{3}+ \beta_{4}(t)X_{4}+0.8X_{5} -1.2 X_{6} + 0.7X_{7}-1.1 X_{8}+\epsilon

where \epsilon\sim N(0,1), \alpha_{0}(t)=2+\sin(2\pi t), \beta_{1}(t)=2.5\exp(2.5t-1) ,\beta_{2}(t)=3t^2-2t+2,\beta_{3}(t)=-4t^3+3 and \beta_{4}(t)=3-2t

See Also

Blend

Examples

data(dat)
length(y)
dim(x)
length(t)
length(J)
print(t)
print(J)
print(kn)
print(degree)

Blend documentation built on April 3, 2025, 10:36 p.m.